Let $G$ be a locally compact group. The question of whether
${\cal H}^1(L^1(G),M(G))$, the first Hochschild cohomology group
of $L^1(G)$ with coefficients in $M(G)$, is zero was first
studied by B. E. Johnson and initiated his development of
the theory of amenable Banach algebras. He was able to show that
${\cal H}^1(L^1(G),M(G)) = \{ 0 \}$
whenever $G$ is amenable, a $[SIN]$-group, or a matrix group
satisfying certain conditions. No group such that
${\cal H}^1(L^1(G),M(G)) \neq \{ 0 \}$
is known. In this paper, we approach the problem of whether
${\cal H}^1(L^1(G),M(G)) = \{ 0 \}$ from several angles.
Using weakly almost periodic functions, we show that ${\cal H}^1(L^1(G),L^1(G))$
is always Hausdorff for unimodular $G$. We also show
that for $[IN]$-groups, every derivation
$D \colon L^1(G) \to L^1(G)$ is implemented,
not necessarily by an element of $M(G)$, but at least by
an element of $\mbox{VN}(G)$, the group von Neumann algebra
of $G$. This applies, in particular, to the group
$G := {\mathbb T}^2 \rtimes \mbox{SL}(2, {\mathbb Z})$,
for which it is unknown whether
${\cal H}^1(L^1(G),M(G)) = \{ 0 \}$. Finally, we analyse
the structure of derivations on $L^1(G)$; an important r\^ole
is played by the closed normal subgroup $N$ of $G$ generated
by the elements of $G$ with relatively compact conjugacy classes.
We can write an arbitrary derivation
$D \colon L^1(G) \to L^1(G)$ as a sum $D = D_N + D_{N^\perp}$,
where $D_N$ and $D_{N^\perp}$ can be tackled with different
techniques. Under suitable conditions, all satisfied by
${\mathbb T}^2 \rtimes \mbox{SL}(2,{\mathbb Z})$, we can
show that $D_N$ is implemented by an element of $\mbox{VN}(G)$
and that $D_{N^\perp}$ is implemented by a measure. 1991 Mathematics Subject Classification:
22D05, 22D25, 43A10, 43A20, 46H25, 46L10, 46M20, 47B47, 47B48.